Optimal designs for the Michaelis–Menten model with correlated observations

In this paper we investigate the problem of designing experiments for generalized least-squares analysis in the Michaelis–Menten model. We study the structure of exact D-optimal designs in a model with an autoregressive error structure. Explicit results for locally D-optimal designs are derived for the case where two observations can be taken per subject. Additionally standardized maximin D-optimal designs are obtained in this case. The results illustrate the enormous difficulties to find exact optimal designs explicitly for nonlinear regression models with correlated observations.     

Locally and maximin optimal designs for multi-factor nonlinear models

This paper considers the search for locally and maximin optimal designs for multi-factor nonlinear models from optimal designs for sub-models of a lower dimension. In particular, sufficient conditions are given so that maximin D-optimal designs for additive multi-factor nonlinear models can be built from maximin D-optimal designs for their sub-models with a single factor. Some examples of application are models involving exponential decay in several variables.     

A Lower Bound to the Measure of Optimality for Main Effect Plans in the Symmetric Factorial Experiments

A design d is called D-optimal if it maximizes det(M _d ), and is called MS-optimal if it maximizes tr(M _d ) and minimizes tr[(M _d )²] among those which maximize tr(M _d ), where M _d stands for the information matrix produced from d under a given model. In this article, we establish a lower bound for tr[(M _d )²] with respect to a main effects model, where d is an s-level symmetric orthogonal array of strength at least one. Non isomorphic two level MS-optimal orthogonal arrays of strength one with N = 10, 14, and 18 runs, non isomorphic three level MS-optimal orthogonal arrays of strength one with N = 6, 12, and 15 runs and non isomorphic four level MS-optimal orthogonal arrays of strength one with N = 12 runs are also presented.     

Optimal Designs for 2 k Choice Experiments

Abstract

In this article we establish the choice sets in the D-optimal design for a choice experiment for testing main effects and for testing main effects and two-factor interactions, when there are k attributes, each with two levels, for choice set size m. We also give a method to construct optimal and near-optimal designs with small numbers of choice sets.     

D-Optimal Designs for Covariate Parameters in Block Design Set Up

The problem considered is that of finding D-optimal design for the estimation of covariate parameters and the treatment and block contrasts in a block design set up in the presence of non stochastic controllable covariates, when N = 2(mod 4), N being the total number of observations. It is clear that when N ≠ 0 (mod 4), it is not possible to find designs attaining minimum variance for the estimated covariate parameters. Conditions for D-optimum designs for the estimation of covariate parameters were established when each of the covariates belongs to the interval [?1, 1]. Some constructions of D-optimal design have been provided for symmetric balanced incomplete block design (SBIBD) with parameters b = v, r = k = v ? 1, λ =v ? 2 when k = 2 (mod 4) and b is an odd integer.     

A catalogue of designs for partial profiles in paired comparison experiments with three groups of factors

A common strategy for avoiding information overload in multi-factor paired comparison experiments is to employ pairs of options which have different levels for only some of the factors in a study. For the practically important case where the factors fall into three groups such that all factors within a group have the same number of levels and where one is only interested in estimating the main effects, a comprehensive catalogue of D-optimal approximate designs is presented. These optimal designs use at most three different types of pairs and have a block diagonal information matrix.     

The Mean Squared Error Optimum Design Criterion for Parameter Estimation in Nonlinear Regression Models

In the context of nonlinear regression models, we propose an optimal experimental design criterion for estimating the parameters that account for the intrinsic and parameter-effects nonlinearity. The optimal design criterion proposed in this article minimizes the determinant of the mean squared error matrix of the parameter estimator that is quadratically approximated using the curvature array. The design criterion reduces to the D-optimal design criterion if there are no intrinsic and parameter-effects nonlinearity in the model, and depends on the scale parameter estimator and on the reparameterization used. Some examples, using a well known nonlinear kinetics model, demonstrate the application of the proposed criterion to nonsequential design of experiments as compared with the D-optimal criterion.     

Modified Robust Second-Order Slope-Rotatable Designs

Generally it is very difficult to construct robust slope-rotatable designs along axial directions. Present paper focuses on modified second-order slope-rotatable designs (SOSRDs) with correlated errors. Modified robust second-order slope-rotatability conditions are derived for a general variance–covariance structure of errors. These conditions get simplified for intraclass correlation structure. A few robust second-order slope-rotatable designs (over all directions, or with equal maximum directional variance slope, or D-optimal slope) are examined with respect to modified robust slope-rotatability. It is observed that robust second-order slope-rotatable designs over all directions, or with equal maximum directional variance slope, or D-optimal slope are not generally modified robust second-order slope-rotatable designs.     

Bayesian D-optimal Accelerated Life Test plans for series systems with competing exponential causes of failure

This paper provides methods of obtaining Bayesian D-optimal Accelerated Life Test (ALT) plans for series systems with independent exponential component lives under the Type-I censoring scheme. Two different Bayesian D-optimality design criteria are considered. For both the criteria, first optimal designs for a given number of experimental points are found by solving a finite-dimensional constrained optimization problem. Next, the global optimality of such an ALT plan is ensured by applying the General Equivalence Theorem. A detailed sensitivity analysis is also carried out to investigate the effect of different planning inputs on the resulting optimal ALT plans. Furthermore, these Bayesian optimal plans are also compared with the corresponding (frequentist) locally D-optimal ALT plans.     

Optimal designs for the logit and probit models for binary data

We consider optimal designs for a class of symmetric models for binary data which includes the common probit and logit models. We show that for a large group of optimality criteria which includes the main ones in the literature (e.g. A-, D-, E-, F- and G-optimality) the optimal design for our class of models is a two-point design with support points symmetrically placed about the ED50 but with possibly unequal weighting. We demonstrate how one can further reduce the problem to a one-variable optimization by characterizing various of the common criteria. We also use the results to demonstrate major qualitative differences between the F - and c-optimal designs, two design criteria which have similar motivation.     

Algorithmic construction of optimal block designs for two-colour cDNA microarray experiments using the linear mixed effects model

ABSTRACT

In this study, methods for efficient construction of A-, MV-, D- and E-optimal or near-optimal block designs for two-colour cDNA microarray experiments with array as the block effect are considered. Two algorithms, namely the array exchange and treatment exchange algorithms together with the complete enumeration technique are introduced. For large numbers of arrays or treatments or both, the complete enumeration method is highly computer intensive. The treatment exchange algorithm computes the optimal or near-optimal designs faster than the array exchange algorithm. The two methods however produce optimal or near-optimal designs with the same efficiency under the four optimality criteria.     

Optimal designs for non-competitive enzyme inhibition kinetic models

In this paper, we present a new method for determining optimal designs for enzyme inhibition kinetic models, which are used to model the influence of the concentration of a substrate and an inhibition on the velocity of a reaction. The approach uses a nonlinear transformation of the vector of predictors such that the model in the new coordinates is given by an incomplete response surface model. Although there exist no explicit solutions of the optimal design problem for incomplete response surface models so far, the corresponding design problem in the new coordinates is substantially more transparent, such that explicit or numerical solutions can be determined more easily. The designs for the original problem can finally be found by an inverse transformation of the optimal designs determined for the response surface model. We illustrate the method determining explicit solutions for the D-optimal design and for the optimal design problem for estimating the individual coefficients in a non-competitive enzyme inhibition kinetic model.     

Highly Efficient Designs for Logistic Models with Categorical Variables

Optimal designs for logistic models generally require prior information about the values of the regression parameters. However, experimenters usually do not have full knowledge of these parameters. We propose a design that is D-optimal on a restricted design region. This design assigns an equal weight to design points that contain more information and ignores those design points that contain less information about the regression parameters. The design can be constructed in practice by means of the rank order of the outcome variances. A numerical study compares the proposed design with the D-optimal and completely balanced designs in terms of efficiency.     

Minimax Design for the Difference Between Estimated Responses for the Quadratic Model Over Hypercubic Regions

Abstract

A second-order model involving the intercept and only the pure quadratic terms is considered for regression over hypercubes. Minimization of the variance of the difference between estimated responses at two points, maximized over all pairs of points in the region of interest in factor space, is taken as the design criterion. Optimal design under the minimax criterion is derived and found to be the one which is also simultaneously A-, D-, and E-optimal for the parameters excluding the intercept. The minimax design is compared with other standard designs and is found to perform extremely well.     

D-Optimal Two-Level Fractional Factorial Designs of Resolution III with Two Dispersion Factors

The problem of finding D-optimal designs, with two dispersion factors, for the estimation of all location main effects is investigated in the class of regular unreplicated two-level fractional factorial designs of resolution III. Designs having length three words involving both of the dispersion factors in the defining relation are shown to be inferior in terms of D-optimality. Tables of factors that are named as the two dispersion factors so that the resulting design is either D-optimal or has the largest determinant of the information matrix are provided. Rank-order of designs is studied when the number of length three words involving either one of the dispersion factors and the number of length four words involving both of the dispersion factors are fixed. Rank-order of designs when the numbers of aforementioned words are less than or equal to ten is given.     

Optimal blocked main effects plans

In this paper, we consider the construction of optimal blocked main effects designs where m two-level factors are to be studied in N runs which are partitioned into b blocks of equal size. For N ≡ 2±od4 sufficient conditions are derived for a design to be Φ _f optimal among all designs having main effects occurring equally often at their high and low levels within blocks and then this result is extended to the class of all designs for the case when the block size is two. Methods of constructing designs satisfying the sufficient conditions derived are also given.     

D-Optimal Axial Designs for Quadratic and Cubic Additive Mixture Models

The paper discusses D-optimal axial designs for the additive quadratic and cubic mixture models σ_1≤i≤q(β_ix_i + β_iix²_i) and σ_1≤i≤q(β_ix_i + β_iix²_i + β_iiix³_i), where x_i≥ 0, x₁ + . . . + x_q = 1. For the quadratic model, a saturated symmetric axial design is used, in which support points are of the form (x₁, . . . , x_q) = [1 ? (q?1)δ_i, δ_i, . . . , δ_i], where i = 1, 2 and 0 ≤δ₂ <_{δ1 ≤} 1/(q ?1). It is proved that when 3 ≤q≤ 6, the above design is D-optimal if δ₂ = 0 and δ₁ = 1/(q?1), and when q≥ 7 it is D-optimal if δ₂ = 0 and δ₁ = [5q?1 ? (9q²?10q + 1)^1/2]/(4q²). Similar results exist for the cubic model, with support points of the form (x₁, . . . , x_q) = [1 ? (q?1)δ_i, δ_i, . . . , δ_i], where i = 1, 2, 3 and 0 = δ₃ <_δ2 < δ_{1 ≤}1/(q?1). The saturated D-optimal axial design and D-optimal design for the quadratic model are compared in terms of their efficiency and uniformity.     

A method for constructing supersaturated designs and its Es2 optimality

A lower bound for the Es² value of an arbitrary supersaturated design is derived. A general method for constructing supersaturated designs is proposed and shown to produce designs with n runs and m = k(n — 1) factors that achieve the lower bound for Es² and are thus optimal with respect to the Es² criterion. Within the class of designs given by the construction method, further discrimination can be made by minimizing the pairwise correlations and using the generalized D and A criteria proposed by Wu (1993). Efficient designs of 12, 16, 20 and 24 runs are constructed by following this approach.     

Robust first-order rotatable lifetime improvement experimental designs

Experimental designs are widely used in predicting the optimal operating conditions of the process parameters in lifetime improvement experiments. The most commonly observed lifetime distributions are log-normal, exponential, gamma and Weibull. In the present article, invariant robust first-order rotatable designs are derived for autocorrelated lifetime responses having log-normal, exponential, gamma and Weibull distributions. In the process, robust first-order D-optimal and rotatable conditions have been derived under these situations. For these lifetime distributions with correlated errors, it is shown that robust first-order D-optimal designs are always robust rotatable but the converse is not true. Moreover, it is observed that robust first-order D-optimal and rotatable designs depend on the respective error variance–covariance structure but are independent from these considered lifetime response distributions.     

Addition of runs to a two-level supersaturated design

The purpose of this article is to introduce a new class of extended E(s²)-optimal two level supersaturated designs obtained by adding runs to an existing E(s²)-optimal two level supersaturated design. The extended design is a union of two optimal SSDs belonging to different classes. New lower bound to E(s²) has been obtained for the extended supersaturated designs. Some examples and a small catalogue of E(s²)-optimal SSDs are also included.